Question

A physical quantity is given by $X=\left[M^a{L}^b{T}^c\right]$. The percentage error in measurement of $M,L$ and $T$ are $\alpha ,\beta$ and $\gamma$ respectively. Then, the maximum % error in the quantity $X$ is:

• A. $a\alpha +b\beta +c\gamma$
• B. $a\alpha +b\beta -c\gamma$
• C. $\frac{a}{\alpha }+\frac{b}{\beta }+\frac{c}{\gamma }$
• D. None of these

Answer 1

The correct option is A $a\alpha +b\beta +c\gamma$

The quantity is given as $X=\left[M^a{L}^b{T}^c\right]$

Taking logrithm on both sides $\ln X=a\ln M+b\ln L+c\ln T$

Differentiating and multiplying by $100$, we get

$\frac{\Delta X}{X}\times 100=a\frac{\Delta M}{M}\times 100$ $+b\frac{\Delta L}{L}\times 100+c\frac{\Delta T}{T}\times 100$ ........(1)

Given: $\frac{\Delta M}{M}\times 100=\alpha$ $\frac{\Delta L}{L}\times 100=\beta$ $\frac{\Delta T}{T}\times 100=\gamma$

Thus % error in $X$, $\frac{\Delta X}{X}\times 100=a\alpha +b\beta +c\gamma$